Common fixed point results for weak contractive mappings in ordered b-dislocated metric spaces with applications
© Hussain et al.; licensee Springer. 2013
Received: 6 July 2013
Accepted: 13 September 2013
Published: 7 November 2013
We first introduce a new concept of b-dislocated metric space as a generalization of dislocated metric space and analyze different properties of such spaces. A fundamental result for the convergence of sequences in b-dislocated metric spaces is established and is employed to prove some common fixed point results for four mappings satisfying the generalized weak contractive condition in partially ordered b-dislocated metric spaces. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results.
1 Introduction and preliminaries
The Banach contraction principle is one of the simplest and most applicable results of metric fixed point theory. It is a popular tool for proving the existence of solution of problems in different fields of mathematics. There are several generalizations of the Banach contraction principle in literature on metric fixed point theory [1–10]. Hitzler and Seda  introduced the concept of dislocated topologies and named their corresponding generalized metric a dislocated metric. They have also established a fixed point theorem in complete dislocated metric spaces to generalize the celebrated Banach contraction principle. The notion of dislocated topologies has useful applications in the context of logic programming semantics (see ). Further useful results can be seen in [13–23].
Definition 1.1 
If , then ;
The pair is called a dislocated metric space or a -metric space. Note that when , may not be 0.
Example 1.2 If , then defines a dislocated metric on X.
Definition 1.3 
A sequence in a -metric space is called: (1) a Cauchy sequence if, given , there exists such that for all , we have or , (2) convergent with respect to if there exists such that as . In this case, x is called the limit of and we write .
A -metric space X is called complete if every Cauchy sequence in X converges to a point in X.
Definition 1.4 A nonempty set X is called an ordered dislocated metric space if it is equipped with a partial ordering ⪯ and there exists a dislocated metric on X.
Definition 1.5 Let be a partially ordered set. Then are called comparable if or holds.
Definition 1.6 
Let be a partially ordered set. A self-mapping f on X is called dominating if for each x in X.
Example 1.7 
Let be endowed with the usual ordering, and let be defined by . Since for all , therefore f is a dominating map.
Definition 1.8 
Let be a partially ordered set. A self-mapping f on X is called dominated if for each x in X.
Example 1.9 
Let be endowed with the usual ordering, and let be defined by for some . Since for all , therefore f is a dominated map.
In the following, we give the definition of a b-dislocated metric space.
Definition 1.10 Let X be a nonempty set. A mapping is called a b-dislocated metric (or simply -metric) if the following conditions hold for any and :
() If , then ;
The pair is called a b-dislocated metric space or a -metric space. It should be noted that the class of -metric spaces is effectively larger than that of -metric spaces, since a -metric is a -metric when .
Here, we present an example to show that in general a b-dislocated metric need not be a -metric.
Example 1.11 Let be a dislocated metric space, and , where is a real number. We show that is a b-dislocated metric with .
Obviously, conditions () and () of Definition 1.10 are satisfied.
So, condition () of Definition 1.10 is also satisfied and is a -metric.
However, if is a dislocated metric space, then is not necessarily a dislocated metric space. For example, if is the set of real numbers, then is a dislocated metric, and is a b-dislocated metric on ℝ with , but not a dislocated metric on ℝ.
Recently, Sarma and Kumari  established the existence of a topology induced by a dislocated metric which is metrizable with a family of sets as a base, where for all and . Also, is a closed ball.
Definition 1.12 We say that a net in X converges to x in and write if .
Note that the limit of a net in is unique. For , we write .
Proof To prove (i), (ii) and (iii), we refer to . To prove (iv), let . Suppose that for each α in Δ, is a net in A such that . Thus, for each positive integer i, there is such that , and such that . Take for each i, then is a directed set if , and . This implies that . □
As a corollary, we have the following.
Consequently, we have the following.
Theorem 1.15 Let ϒ be the family of all subsets A of X for which and are the complements of members of ϒ. Then the is a topology for X and the -closure of a subset A of X is .
Definition 1.16 The topology obtained in Theorem 1.15 is called the topology induced by and simply referred to as the -topology of X; and it is denoted by .
Now we state some propositions and corollaries in which can be proved following similar arguments to those given in .
Proposition 1.17 Let . Then iff for every , .
Corollary 1.18 or , .
Corollary 1.19 A set is open in if and only if for every , there is such that .
Proposition 1.20 If and , then is an open set in .
Corollary 1.21 If and for , then the collection is an open base at x in . If is a b-metric and , then coincides with the metric topology.
Proposition 1.22 is a Hausdorff space.
Proof If and , then . □
Corollary 1.23 If , then the collection is an open base at x for . Hence, is first countable.
Remark 1.24 The above corollary enables us to deal with sequences instead of nets.
Motivated by Proposition 3.2 in , we have the following proposition for the b-dislocated metric space.
For all , we have .
is a b-metric.
For all and all , we have .
Proof We show that (iii) implies (i). Since for all , there exists some with . But for all , we have . Therefore, for all . Hence, . □
If is a b-dislocated metric space, then , where is a b-metric space. Indeed, is a b-dislocated metric space, so assertion now follows immediately from the above proposition.
Definition 1.26 A sequence in a b-dislocated metric space converges with respect to (-convergent) if there exists such that converges to 0 as . In this case, x is called the limit of , and we write .
Proposition 1.27 Limit of a convergent sequence in a b-dislocated metric space is unique.
Proof Let x and y be limits of the sequence . By properties () and () of Definition 1.10, it follows that . Hence, , and by property () of Definition 1.10 it follows that . □
Definition 1.28 A sequence in a b-dislocated metric space is called a -Cauchy sequence if, given , there exits such that for all , we have or .
Proposition 1.29 Every convergent sequence in a b-dislocated space is -Cauchy.
Proof Let be a sequence which converges to some x, and . Then there exists with for all . For , we obtain . Hence, is -Cauchy. □
Definition 1.30 A b-dislocated metric space is called complete if every -Cauchy sequence in X is -convergent.
The following example shows that in general a b-dislocated metric is not continuous.
that is, , but as .
We need the following simple lemma about the -convergent sequences in the proof of our main results.
In particular, if , then we have .
Taking the lower limit as in the first inequality and the upper limit as in the second inequality, the result follows. Similarly, using again the triangle inequality, the last assertion follows. □
Definition 1.33 
Let f and g be two self-maps on a nonempty set X. If , for some x in X, then x is called a coincidence point of f and g, where w is called a point of coincidence of f and g.
Definition 1.34 
Let f and g be two self-maps defined on a set X. Then f and g are said to be weakly compatible if they commute at every coincidence point.
Definition 1.35 Let be a b-dislocated metric space. Then the pair is said to be compatible if and only if , whenever is a sequence in X so that for some .
2 Common fixed point results
and . If for every non-increasing sequence and a sequence with , for all n such that , we have and either
(a1) are compatible, f or S is continuous and is weakly compatible, or
(a2) are compatible, g or T is continuous and is weakly compatible,
then f, g, S and T have a common fixed point. Moreover, the set of common fixed points of f, g, S and T is well ordered if and only if f, g, S and T have one and only one common fixed point.
which gives and so , which further implies that . Thus, becomes a constant sequence, hence, is a Cauchy sequence.
which yields that , or, equivalently, , a contradiction.
Now, we show that y is a common fixed point of f, g, S and T.
which gives , or, equivalently, .
which implies that , so .
which implies that , so we have . Therefore, .
The proof is similar when f is continuous.
Similarly, if (a2) holds, then the result follows.
So, we have , a contradiction. Therefore . The converse is obvious. □
In the following theorem, we omit the continuity assumption of f, g, T and S and replace the compatibility of the pairs and by weak compatibility of the pairs, and we show that f, g, S and T have a common fixed point on X.
and . If for every non-increasing sequence and a sequence with , for all n such that , we have , and the pairs and are weakly compatible, then f, g, S and T have a common fixed point. Moreover, the set of common fixed points of f, g, S and T is well ordered if and only if f, g, S and T have one and only one common fixed point.
Now we prove that v is a coincidence point of f and S.
which implies that , so from (2.28) we obtain .
As f and S are weakly compatible, we have . Thus, y is a coincidence point of f and S.
which implies that , so we have . Therefore, .
Now, similar to the proof of Theorem 2.1, indeed from (2.20)-(2.22), we have . Therefore, , as required. The last conclusion follows similarly as in the proof of Theorem 2.1. □
Now, we give an example to support our result.
Thus, f, g, S and T satisfy all the conditions of Theorem 2.1. Moreover, 0 is a unique common fixed point of f, g, S and T.
and . If for every non-increasing sequence and a sequence with , for all n such that , we have , then f and g have a common fixed point. Moreover, the set of common fixed points of f and g is well ordered if and only if f and g have one and only one common fixed point.
Proof Taking S and T as identity maps on X, the result follows from Theorem 2.2. □
and . If for every non-increasing sequence and a sequence with , for all n such that , it implies that , then f and g have a common fixed point. Moreover, the set of common fixed points of f and g is well ordered if and only if f and g have one and only one common fixed point.
Proof If we take S and T as the identity maps on X and for all , then from Theorem 2.2 it follows that f and g have a common fixed point. □
Remark 2.6 As corollaries we can state partial metric space as well as b-metric space versions of our proved results in a similar way, which extends recent results in these settings.
3 Existence of a common solution for a system of integral equations
where . The purpose of this section is to present an existence theorem for a solution to (3.1) that belongs to (the set of continuous real functions defined on ) by using the obtained result in Corollary 2.4.
Here, . The considered problem can be reformulated in the following manner.
for all and for all .
for all , is a complete b-dislocated metric space with .
for all . Moreover, in , it is proved that is regular.
Now, we will prove the following result.
- (ii)for all and , we have
- (iii)for all and with , we have
Then the integral equations (3.1) have a common solution .
Proof From condition (ii), f and g are dominated self-maps on X.
Let with .
Taking and in Corollary 2.4, there exists , a common fixed point of f and g, that is, x is a solution for (3.1). □
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.
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